Another point to be made is that CC never said to me that the goal was to eliminate movement: only to reduce it to a minimum. Reducing it to a level such that it doesn't get in the way when performing music.

The perception of movement is similar to pressure; it's not a flaw. It's not a bogeyman. If it doesn't get in the way, don't worry about it.

In addition to the things others have mentioned about the six notes, I've found that removing unnecessary movements when I start to play (I play the six notes after the first Clarke exercise every day) has helped to clean up my playing. Starting off with a good feel for lip position gives me more confidence in pitching notes. It might be similar to how those doing martial arts often start with simple building block moves.